4 ms·
I guess I'm not totally surprised. Could you give a reference? I'm not sure what this would mean exactly or what the utility would be.
by sfpotter 4y ago
I guess I'm not totally surprised. Could you give a reference? I'm not sure what this would mean exactly or what the utility would be.
- enriquto 4y agoI guess the original article [0] is a good starting reference. A post [1] in Tao's blog gives a nice overview. If you just want a short statement of the discrete uncertainty principle: If you define the discrete Fourier transform F(u) for functions u defined on a finite abelian group G, and you denote by |S(u)| the cardinal of the support of u, then you have the inequalities: |S(u)| · |S(F(u))| >= |G| and (as consequence) |S(u)| + |S(F(u))| >= 2 sqrt(|G|) Notice that this contains as a particular case the discrete Fourier transform, where the abelian group in question is the integers modulo N. This has a very practical and intutive interpretation: if the signal u is very localized, then its spectrum F(u) cannot be very localized at the same time, for their supports must be large enough (with respect to the total size of the domain). [0] Donoho D. L. and Stark P. B., Uncertainty Principles and Signal Recovery, SIAM Journal of Applied Mathematics, 49 (1989), 906–931 [1] https://terrytao.wordpress.com/2010/06/25/the-uncertainty-principle/ https://terrytao.wordpress.com/2010/06/25/the-uncertainty-pr...
- sfpotter 4y agoAh, very cool! Yeah, that makes sense. Appreciate you taking the time to write up a quick explanation and send a reference. I'll check it out.